Global existence and asymptotic behavior of solutions for nonlinear parabolic equations on unbounded domains

We study the existence and the asymptotic behavior of positive solutions for the parabolic equation on D×(0,∞), where is a some unbounded domain in and V belongs to a new parabolic class J^∞ of singular potentials generalizing the well-known parabolic Kato class at infinity P^∞ introduced recently by Zhang. We also show that the choice of this class is essentially optimal.     

Boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms

In this paper, we study the boundary behavior of solutions to boundary blow-up elliptic problems , where Ω is a bounded domain with smooth boundary in R^N, q>0, , which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary, and f is rapidly varying or normalized regularly varying at infinity.     

Inhomogeneous infinity Laplace equation

We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation in domains in Rn. We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition f does not change its sign. We also discover a characteristic property, which we call the comparison with standard functions property, of the viscosity sub- and super-solutions of the equation with constant right-hand side. Applying these results and properties, we prove the stability of the inhomogeneous infinity Laplace equation with nonvanishing right-hand side, which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right-hand side and boundary data are perturbed. In the end, we prove the stability of the well-known homogeneous infinity Laplace equation , which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right-hand side and boundary data are perturbed simultaneously.     

Locating the peaks of the least energy solutions to an ellyptic system with Neumann boundary conditions

We consider a system of the form in Ω with Neumann boundary condition on ∂Ω, where Ω is a smooth bounded domain in and f,g are power-type nonlinearities having superlinear and subcritical growth at infinity. We prove that the least energy solutions to such a system concentrate, as ε goes to zero, at a point of the boundary which maximizes the mean curvature of the boundary of Ω.     

Second-order semilinear elliptic inequalities in exterior domains

We study the problem of existence and nonexistence of positive solutions of the semilinear elliptic inequalities in divergence form with measurable coefficients in exterior domains in . For W(x)?|x|^−σ at infinity we compute the critical line on the plane (p,σ), which separates the domains of existence and nonexistence, and reveal the class of potentials V that preserves the critical line. Example are provided showing that the class of potentials is maximal possible, in certain sense. The case of (p,σ) on the critical line has also been studied.     

Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity

We study smoothing properties for time-dependent Schrödinger equations , , with potentials which satisfy V(x)=O(|x|^m) at infinity, m?2. We show that the solution u(t,x) is 1/m times differentiable with respect to x at almost all , and explain that this is the result of the fact that the sojourn time of classical particles with energy λ in arbitrary compact set is less than CTλ^−1/m during [0,T] when λ is very large. We also show Strichartz's inequality with derivative loss for such potentials and give its application to nonlinear Schrödinger equations.     

On sign-changing and multiple solutions of the p-Laplacian

In this paper, we construct the pseudo-gradient vector field in , by which we obtain the positive and negative cones of are both invariant sets of the descent flow of the corresponding functional. Then we use differential equations theory in Banach spaces and dynamics theory to study p-Laplacian boundary value problems with “jumping” nonlinearities at zero or infinity, and get new multiple solutions and sign-changing solutions theorems of p-Laplacian.     

Connectivity preserving transformations for higher dimensional binary images

An N-dimensional digital binary image (I) is a function I:Z^N→{0,1}. I is connected if and only if its black pixels and white pixels are each (3^N−1)-connected. I is only connected if and only if its black pixels are (3^N−1)-connected. For a 3-D binary image, the respective connectivity models are and . A pair of (3^N−1)-neighboring opposite-valued pixels is called interchangeable in a N-D binary image I, if reversing their values preserves the original connectedness. We call such an interchange to be a (3^N−1)-local interchange. Under the above connectivity models, we show that given two binary images of n pixels/voxels each, we can transform one to the other using a sequence of (3^N−1)-local interchanges. The specific results are as follows. Any two -connected 3-dimensional images I and J each having n black voxels are transformable using a sequence of O((c₁+c₂)n²) 26-local interchanges. Here, c₁ and c₂ are the total number of 8-connected components in all 2-dimensional layers of I and J respectively. We also show bounds on connectivity under a different interchange model as proposed in [A. Dumitrescu, J. Pach, Pushing squares around, Graphs and Combinatorics 22 (1) (2006) 37-50]. Next, we show that any two simply connected images under the , connectivity model and each having n black voxels are transformable using a sequence of O(n²) 26-local interchanges. We generalize this result to show that any two , -connected N-dimensional simply connected images each having n black pixels are transformable using a sequence of O(Nn²)(3^N−1)-local interchanges, where N>1.     

Existence of sign-changing solutions for semilinear singular elliptic equations with critical Sobolev exponents

We consider the semilinear elliptic problem in Ω, u=0 on ∂Ω, where 0∈Ω is a smooth bounded domain in RN, N?4, , is the critical Sobolev exponent, f(x,⋅) has subcritical growth at infinity, K(x)>0 is continuous. We prove the existence of sign-changing solutions under different assumptions when Ω is a usual domain and a symmetric domain, respectively.     

Bubble towers for supercritical semilinear elliptic equations

We construct positive solutions of the semilinear elliptic problem with Dirichet boundary conditions, in a bounded smooth domain Ω⊂R^N(N?4), when the exponent p is supercritical and close enough to and the parameter λ∈R is small enough. As , the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. Our result extends the result of Del Pino et al. (J. Differential Equations 193(2) (2003) 280) when Ω is a ball and the solutions are radially symmetric.     

Classical and non-classical solutions of a prescribed curvature equation

We discuss existence and multiplicity of positive solutions of the one-dimensional prescribed curvature problem

depending on the behaviour at the origin and at infinity of the potential . Besides solutions in W^2,1(0,1), we also consider solutions in which are possibly discontinuous at the endpoints of [0,1]. Our approach is essentially variational and is based on a regularization of the action functional associated with the curvature problem.     

L stability estimate for a one-dimensional Boltzmann equation with inelastic collisions

In this paper, we study the L¹ stability of a one-dimensional Boltzmann equation on the line with inelastic collisions in Rend. Sem. Mat. Fis. Milano 67 (1997) 169-179. Under the suitable assumptions on the initial data, we construct a nonlinear functional which measures L¹ distance between two mild solutions, and is nonincreasing in time t. Using the time-decay estimate of , we show that mild solutions are L¹-stable:

     

Asymptotic stability at infinity for differentiable vector fields of the plane

Let be a differentiable (but not necessarily C¹) vector field, where σ>0 and . Denote by R(z) the real part of z∈C. If for some ?>0 and for all , no eigenvalue of DpX belongs to , then: (a) for all , there is a unique positive semi-trajectory of X starting at p; (b) it is associated to X, a well-defined number I(X) of the extended real line [−∞,∞) (called the index of X at infinity) such that for some constant vector v∈R² the following is satisfied: if I(X) is less than zero (respectively greater or equal to zero), then the point at infinity ∞ of the Riemann sphere R²∪{∞} is a repellor (respectively an attractor) of the vector field X+v.     

On strong convergence to 3D steady vortex sheets

In this paper, we first establish a strong convergence criterion of approximate solutions for the 3D steady incompressible Euler equations. For axisymmetric flows, under the assumption that the vorticity is of one sign and uniformly bounded in L¹ space, we obtain a sufficient and necessary condition for the strong convergence in of approximate solutions. Furthermore, for one-sign and L¹-bounded vorticity, it is shown that if a sequence of approximate solutions concentrates at an isolated point in (r,z)-plane, then the concentration point can appear neither in the region near the axis (including the symmetry axis itself) nor in the region far away from the axis. Finally, we present an example of approximates solutions which converge strongly in by using Hill's spherical vortex.